Wave motion in low-pressure-ratio rectangular and pyramidal shock tubes

December

1975

WAVE IDl'ION IN LOW-PRESSURE-RATIO

RECTANGULAR ANI> PYRAMIDAL SHOCK

TUBES

by

tECHNISCHE:

HOGESG WOL ti - :-1 WCHTVAART-EN RUIP!/TEVtARTTECt! .. .:i\

BIBliOTME1i:C(

Kluyve

rwe9

1 OELFT

James Joseph Gottlieb

urIAS

Report No.

199

ON ISSN 0082-5255

..

WAVE MOTION IN LOW-PRESSURE-RATIO RECTANGULAR AND FYRAMIDAL SHOCK TUBES

December, 1975

by

James Joseph Gott1ieb

UTIAS Report No. 199 GN ISSN 0082-5255

Acknowledgements

I should like to thank both Dr. J. H. deLeeuw, Direc~or, and Dr.

I. I. Glass for giving me the opportunity of completing the present research.

Dr. Glass's continual interest and encouragement and critical review of the

manuscript are very much appreciated.

I wish to thank Reinhard Gnoyke for his competent assistance in obtaining the experimental data presented in this report.

The assistance received from Mrs. Mary Fiorellino, Mr. Carlos Basdeo and Mr. John McCormack in typing and printing this report is very much

appreciated.

The financial assistance provided by the Canadian Transportation Development Agency, Canadian Ministry of Transport, the National Research Council of Canada, and the Air Force Office of Scientific Research, Air Force Systems Command, and United States Air Force, under Grant No. 72-2274,

are acknowledged with thanks.

Abstract

Closed-form solutions based on acoustic theory have recently been obtained to describe the wave motion in both low,-pressure-ratio rectangular (constant area) and pyramidal shock "tubes which utilize different driver and channel gases. These new solutions are in excellent agreement with ex-perimental data. This work should be of interest to researchers who are using shock tubes or similar devices to produce impulse noise, in particular the simulated sonic boom, in order to facilitate studies of the effects of impulse sound on humans, animals and structures. Furthermore, this work is relevant to the understanding of the wave motion produced by weak planar and spherical explosions of finite size.

1. 2.

3.

4.

5.

Acknow1edgements Abstract Tab1e of Contents List of Symbo1s INTRODUCTION

RECTANGULAR SHOCK TUBE FYRAMIDAL SHOCK TUBE

3.1

General Solution

3.2

Special Solution

Tab1e of Contents

DISCUSSIONS AND CONCLUSIONS REFERENCES TABLES FIGURES APPENDICES iv ii iii iv v 1 5

15

15

30

33

37

a A h. (~) ~ H(t} m o M p D.p 2 D.p o r r o List of Symbols speed of sound

sound speed of the channel gas sound speed of the driver gas

cross-sectional area of a rectangular shock tube ith velocity potential function of the driver gas

(inward moving waves)

ith velocity potential function of the channel gas (outward moving waves}

ith vel 0 city potential function of the driver gas

(outward moving waves)

Heavjside unit step function; H(t}

=

°

if t

<

0, H(t}

=

1 if t

>

0.

mass of the driver gas molecular weight

molecular weight of the channel gas molecular weight of the driver gas absolute pressure

absolute pressure of the channel gas absolute pressure of the driver gas

överpressure, excess pressure or pressure difference overpressure of the channel gas

overpressure of the driver gas

pressure difference across the diaphragm in a shock tube radial distance

radial location of the diaphragm, length of the pyramidal driver

initial radial location of a fluid particle in the charme]. of a pyramidal shock tube

6.r o

:

R

R o t ',:T T . l' T 2 6.u 6.u 1 6.u 2

v

x x o '1 ₂ fluid-particle displacement (r-r*) contact-surf'ace displacement (r-r ) o gas constant

universal gas constant (R MR) o

time

temperature

temperature of the channel gas temperature of the driver gas fluid velocity

velocity of' the channel gas velocity of' the driver gas vol ume of' tie driver gas

initial volume of the driver gas

distance measured from the closed end of' the rectangular shock tube

location of' the diaphragm in a rectangular shock tube, driver length

initial location of' a fluid particle in the channel of a rectangular shock tube

(a t - x)/x or (~t - r)/r_o

2 0

ratio of the specific heats

specific heats ratio of the channel gas specific heats rat"io of the driver gas (a t + x)/x or (a t,.. r)/r

2 0 2 0

(a₁P₁ + a

2P 2)/a1

(p

2 -

p

i

)

wave length

. . . . - - - -- - - -_._---- -p t

.

(a t - x)/x or (a t - r + r )a₂/a r 1. 0 1. 0 1 . 0 density

density of the char~el gas density of the driver gas velo city potential

velocity potential of the channel gas velocity potential of the driver gas

factorial function (O~

=

1)

binomial coefficients [m~/n~(m - n)~]

1. INTRODUCTION

Noise pollutión is becoming more significant in recent times and is contributing to the stresses affecting man. Ongoing studies into the effects of noise on humans (e.g., see Kyter's book - Ref. 1), animals and even structures are proceeding, fortunately, at a eorresponding accelera-ted pace. One important and aeti ve area of eurrent noi se research is. the study of the effects of impulse noise. A particularly important impulse sound is the sonie boom, whose impact on society is being assessed before supersonic transport (SST) aircraft sueh as tbe Anglo-Freneh Concorde and Soviet TU-144 are introdueed into extensive commercial service.

At the University of Toronto, Institute for Aerospace Studies (UTIAS), three different but eomplementary sonie-boom simulators have been developed to help assess eurr.ent societal problems associated wi th the sonic boom. The first of two major facilities, a loudspeaker-driver booth (Refs. 2 and 3), can easily accommodate one human subj eet or a few

small caged animals in its solidly built and sealed chamber (volume of 2ID3) to facilitate studies of human and animaJ. response to a simulated full-scale' sonic boom. The seeond major faeility (Refs. 2, 3, 4,

5,

and

6),

a travelling-wave horn in the form of a horizontal concrete

pyramid (25 m long, 3-m-square base), has at its apex a speeially designed vaIV'e which regulates tbe air discharge from a large reservoir into the pyramidal horn. This controlled discharge of air generates in the horn

interior a simulated full-scale sonic boom for human, animaJ. or struc-tural response investigations. Alternatively a shock-tube driver can be installed at the horn apex to produce a simulated short-duration sonic boom to facili tate certain response tests. The third sonic- boom

simu-_lator (Refs. 3 and 7), a portable shock tube (11 kgm., 1 m long) having a constant-area driver and an exponential horn, can be easily transpor-ted and operatranspor-ted by one person to conduct wildlife field tests. A

simulated short-duration sonic boom can be produced and directèd at wild-life in their natural hàbitat in order to study their startle response.

In order to illustrate the type of wave that must be produced by a sonie-boom simulator, and also for future referenee, an idealized

overpressure signature of a sonic boom is sketched in Fig. 1. The

more important parameters which are commonly used to describe the various parts of the signature include the peak overpressure, rise time, dura-tion (or wave length) , and wave form which may vary somewhat from the ideal 'N' shape. It is worth noting that respective values of peak over-pressure, duration and rise time are 100

NIm?,

300 ms and 1 ms for a typieal sonic boom from a current SST aircraft and also from large mili-tary bomber supersonic aircraft. In the case of shorter milimili-tary fighter

supersonic aireraft only the duration is significantly d.ifferent, being

eor~espondingly short er at about 100 ms.

The analytical and experimental work given in this report on the wave motion in low-pressure-ratio rectangular and pyramidal shock

-tubes utilizing different driver and channel gases is a natural contin-uation of previous sonic-boom-simulation work at UTIAS. Also, the present work is a direct extension of sonic-boom-simulation research on

state-ments are elaborated on in the following part of this introduction.

A mmlber of geometrically different shock tubes have been constr-ucted and tested by English, French, American and Canadian researchers to assess,·_thèir capability of producing a good 'N-wave' to simulate the sonic boom (Fig. 1). Some of the more important shock tubes are illustrated

schematically in Fig. 2, and the developmental and research efforts made O.y various researchers on such shock tubes are summarized in Table 1. Each

shock tube shown in Fig. 2 has been named according to the geometrical shape Ç>f its driver and channel. The adjectives 'rectangular' and 'pyramidal' arise quite naturally owing to the geometry of the driver and channel of shock tubes commonly used in the laboratory. More generally, however, in this report rectangular will refer to any duct where the cross-sectional area is constant with distance (e .g., cylindrical) , and pyramidal will

refer to any duct where the cross-sectional area incfeases directly with the square of the distance from the center of symmetry (e.g., conical).

For shock tubes cammonly used for sonic-boom simulation purposes the driver and channel gases are both normally air at atmospheric tempe:r>a-ture. The driver air, however, is initially at a slightly higher pressure than the atmospheric pressure air in the channel. Consequently the

pres-sure ratio for the driver and channel gases separated initially by a thin diaphragm is very nearly equal to uni ty. On breaking the diaphragm in such a low-pressure-ratio shock tube the ensuing wave motion results in a very weak shock wave or simulated sonic boom moving in th e channel gas.

Closed-form solutions for the wave motion in low-pressure-ratio rectangular, pyramidal , pyramidal--rectangular, pyramidal-pyramidal and rectangular-pyramidal shock tubes which use identical driver and channel gases have been obtained previously (Refs. 5 and

6).

The rectangular shoèk tube produces a constant-amplitude pulse as illustrated by the overpres-sure signature sketched in Fig. 3a. In the case of the pyramidal shock tube an N-shaped pulse is produced as shown in Fig. 3b. The other pyramidal-rectangular, pyramidal-pyramidal and rectangular-pyramidal shock tubes each produces a distorted N-wave followed by additional waves or disturbances

as illustrated in Fig. 3c, 3d and 3e respectively. Obviously the pyramidal shock' tube is best for producing a good N-wave for sonic-boom simulation purposes. Note th at the rapid pressure rise across both the front and rear

shocks of the N-wave equals 6Poro/2r and the duration of the N-wave is 2ro/a, where the respective symbols 6po' a, rand ro denote the pressure difference across the di aphr agm, sound speed of the driver gas, radial distance measured from the driver apex, and the diaphragm location or driver length.

A large pyramidal shock tube is required if the N-wave produced in the channel is to simulate a full-scale sonic boom from an SST aircraft. Sincë the duration of the N-wave in the shock tube equal:s. 2ro/a and the duration of a simulated sonic boom has to be approximately 300' ms, the cor-respDnding length of the shock-tube (ro) needs to be about 50 m. Owing to inadequate breakage of a large diaphragm in a low-pressure-ratio shock tube (Ref. 12), the largest diaphragm ~aving sufficiently good breaking charac-teristics appears to be about 1 m in area. The maximum divergence angle of a pyramidal shock tube is therefore about one-fiftieth of a radian. Now the interior test section of the pyramidal channel must be at least 3 m on each

..

sideto adequately accommodate a human subject without unduly blocking the path of the simulated sonic boom. For such a test section size and diver-gence angle the resulting length of the pyramidal shock tube is an incredible 150 m. In practice the shock tube is even longer because a reflection

eliminator is required to cover the open end of the channel. For the sake of interest the large Franco-German pyrarnidal shock tube (Ref. 14), which is 189 m in length, is depicted in Fig. 4. Note that different diaphragm

sta-tions have been provided such that the N-wave duration can be varied conveniently from 100 to 300 ms.

Owing to the large size required of a pyramidal shock tube if a full-sca!l..e sonic boom is to be simulated, the initial construction can be

a costly endeavour. If a pyramidal-pyramidal shock tube instead of a pyramidal shock tube could be used, then the shock-tube length and thus the initial con-struction cost might be reduced. The idea behind this scheme is brought out by the diagrams given in Fig. 5. In order to maintain the same sized driver,

diaphragm and test section of a pyrarnidal shock tube (Fig. 5a), the divergence angle of the channel is increased, thereby producing a shorter shock tube (Fig. 5b). This resulting pyramidal-pyramidal shock tube, however, produces a distorted N-wave (Fig. 3d), which is normally undesirable for sonic-boom simulation purposes. In fact it is worth mentioning that, based on the ex-perimental results obtained from prototype pyramidal-pyrarnidal shock tubes (Ref. 12), French and German researchers deemed the distortion in the N-wave as unacceptable, and they consequently constructed the large Franco-German pyramidal shock tube mentioned previously and show'rl in Fig. 4.

Another possible method of achieving a shorter and thus less costly pyrarnidal shock tube for the simulation of a full-scale sonic boom is to use a driver gas which has a low sound speed. (Although the channel gas could be the same as the driver gas, i t would probably be more conven-ient to use air in the channel.) Because the predicted duration of the N-wave produced in the shock tube is 2ro/a, if a gas having a lower sound

speed (a) than air is used in the driver, the correct full-scale duration of

~~onic boom can be accomplished with a shorter-than-normal driver (length ro) änd hence a shorter pyramidal shock tube. For example, the use of an economical gas like carbon dioxide (C02) instead of air in the driver and air in the channel would result in a fairly significant reduction of 2'è'/o in shock-tube length. The use of more exotic and expensive dichlorodifluoro-methane (C C12 F2), sulfur hexafluoride (SF6) or octofluorocyclobutane (C4 F8) would result in an even more significant reduction in shock-tube length by 56%, 60% or 61% respectively.

Another possible advantage of usïng a driver gas having a low sound speed is to improve the capability of an existing low-pressure-ratio pyramidal shock tube. Most of the existing shock tubes have been designed to be ~hort to minimize construction costs, and they consequently produce a short-duration simulated sonic boom. (Such short-duration booms can be used for studies of wave diffraction over and into model buildings, wave propagation over reduced-scale land topologies, and certain animal response tests). Since atmospheric temperature air is normally used in both the driver and channel the standard technique of changing the N-wave duration for new tests is to alter the driver length by changing the diaphragm loc-ation. As existing shock tubes are normally short (less than 15 m long) an upper limit exists on the driver length and therefore on the N-wave duration.

Consequently, the only means of further increasing the N-wave duration and thus extending the capability of existing shock tubes is to use a driver gas which has a sufficiently low sound speed. For example, the N-wave dur-ation can be increased markedly by a factor of

l

:

a,

2.3, 2.5 or 3.0 when the driver gas is C02' CC12F2' SF6 or C4FS, respectively, instead of air. It should be noted th at the N-wave duration can be increased further by lowering the temperature ofthe driver gas. However, this method is not only imprac'ticable but the increase in duration is not very significant, being only 20% for a decrease in driver-gas temperature from 300 to 200 K.

The technique of using a driver gas with a low sound speed in a pyramidal shock tube to increase the N-wave duration both to extend the

capabili ty of an existing shock tube and to investigate the feasibili ty of using different driver and channel gases in a larger sonic-boom simula-tor was originally tested in England (Ref.

9).

Carbon dioxide was used in thet-driver of a small conical shock tube (driver length of 60 cm, channel length of 140 cm) and showed that a good N-wave could be produced in the channel air. The N-wave duration was an expected 1.3 times langer than that for the case when air was used in both the driver and channel. To obtain anc even longer duration N-wave with the same shock tube the use of a special gas having the trade name 'Arcton' was proposed, as its low sound speed is only 43% of that for air. Test results for the case of Arcton as the driver gas were unfortunately not reported.

The British researchers (Ref. 9) did not notice any significant distortiàn in the overpressure signature of the N-wave produced in

their conical shock tube when carbon dioxide was used in the driver and the Channel gas was air. However experiments at UTIAS showed that if the

sound speeds of the driver and channel gases were radically different (e.g., CC12F2, SF6 or C4F8 being the driver gas with air in the channel) then a noticeably distorted N-wave would be produced. This distortion in the N-wave is similar to that in the N-wave produced by a pyramidal-pyra-midal shock tube which used identical driver and channel gases (Fig. 3d). If the undesirable distor'tion in the N-wave is significant then the con-cept of using different driver and channel gases in a pyramidal shock tube either to increase the duration of an N-wave produced in an existing

short shock tube or to enable a short and inexpensive shock tube to pro-duce a simulated full-scale sonic boom is not valid. This will be verified subsequently in this report by analytical and experimental data.

The acoustic analysis for the wave motion in a pyramidal shock tube utilizing different driver and channel gases was found to be quite complex analytically. Prior to obtaining tre acoustic solution for the pyramidal shock tube a similar but simpler acoustic solution was ohtained for the case of a rectangular shock tube utilizing different driver and channel gases. For interest and completeness both of these acoustic analyses are given. The simpler analysis for the rectangular shock tube is presented first in Chapter 2 and the more complex analysis for the pyramidal shock tube follows in Chapter 3. Experimental data for both the rectangular and pyramidal shock tubes are also given, verifying the analyses.

It is worth mentioning th at the respective acoustic analyses for the rectangular and pyramidal shock tubes are applicable directly to weak

planar and spherical explosions of finite size. Consequently the acoustic analysès are of fundament al importance

t

o

the understanding of the wave motion of planar and spherical explosions, for which the explosion gas differs from the ambient gas.

2 • RECTANGULAR SHOCK TUBE

The rectangular shock tube, which is depicted in Fig. 2a, con-sists essentially of a constant cross-sectional area driver and channel which are joined at the diaphragm station. The diaphragm initially sep-arates a normally atmospheric pressure channel gas from a slightly higher pressure driver gas. For generality of the analysis let the initially

quiescent driver and channel gases be not only different but also have dif-ferent temperatures. Before the diaphragm is broken the appropriate ini tial conditions of the driver and channel gases for the analysis can therefore be summarized mathematically as follows.

Channel (x

>

x ): _6Pl = 0 (2.1) 0 6,\ = 0 (2.2) Driver (0

<

x

<

x ): _{~2 6p} (2.3) 0 0 6U 2 = 0 (2.4)

The respective symbols 6p, 6U, and 6po denote overpressure wi th respect to atmospheric pres sure, perturbation flow veloci ty or partiele veloci ty, and pressure difference across the di aphragm , whereas the respective subscripts 1 and 2 refer the appropriate symbols to the channel and driver gases. The distanee x is measured along the shock tube starting from the closed end of of the driver as shown in Fig. 2a, and the diaphragm is located at distance xo, thereby making the driver length equalto xo'

For the preceding initial conditions the ensp.J.ng wave motion in the rectangular shock tube after the diaphragm is broken is rather complex. This wave motion can be depicted conveniently on a time-distance diagram, as shown in Fig.

6,

where the locus of each wave front has been drawn. On breaking the di1_{aphragm in the shock tube the resulting rapidly expanding} driver gas produces a weak shock wave (gl) in the ch~nnel gas. Simultan-eously a weak rarefraction wave (fl) moves into the driver gas and

eventually reaches and reflects from the closed end. This reflected wave (hl) propagates through the driver gas and encounters the contact surface or interface of the driver and channel gases. Because the contact-surface displacement is negligible in( comparison with the driver length, the con-tact-surface path is shown in Fig.

6

simply as a vertical dashed line. Owing to the change in specific impedance (product of the density and sound speed) across the contact surface the reflected wave (hl) is partially

transmitted to the channel (g2) and partially reflected back into the driver (f2). The subsequent oscillatory wave motion between the contact surface and closed end of the driver creates a sequence of waves in the channel. The integrated result of all of these individual waves (gi,:i

=

1,2, ... , (0) yields the total wave in the channel.

For a low-pressure-ratio rectangular shock tube for which the pressure difference across the diaphragm b.Po is inuch 1ess than the absolute pressure of the channe1 gas, the individual and total waves in the driver and channel are re1ative1y weak. The wave motion can therefore be described adequately by using we11-known acoustic theory. The one-.dimensiona1 p1anar waye equations which govern the acoustic wave motion in the channe1 and driver gases are given below.

Channel (x> x ): _o Driver (0

<

x

<

x): o 02<1> 1 ot2 02<1> 2

~

-= 2 0 2 <1> 1 al ox2 2 0 2 <1> ₂ a 2

_~

The respective symbo1s <1>, a and t denote tota1 velocity potential, sound speed and time.

For the analysis it is convenientto express the total velocity potential of the channe1 gas (<1>1) and the driver gas (<1>2) in a more basic form consisting of the sum of velocity potentials of individua1 waves in the channe1 and driver respecti ve1y • The new forms of <1>1 and <1>2, and re1ated expressions for overpressure and partic1e velocity, are sunnnarized below. Channe1 (x > x ): o 00 <I> 1 =

I [

gi ( s) H (s - 2 ( i -1) all a2 + 1}

J

i=l Driver (0

<

x

<

x ): o -/::,p t o ~2 otY

+ _{I [ fi (1l) H(1l- 2i+1 } + hi(t')}

H(t'-2i+~}J

i=l 0<1>2 b.P2

=

-P2

dV

0<1>2 b.u2 =

'

Öx

(2.5) (2.6) (2.7) (2.8)

(2.10)

(2.11) (2.12)

In the above expressions: p denotes density; the respective nondimen-sional variables ~, _{'1, (3 equal (alt - x)/xo '} (~t + x)/xo and (~t - x)/xo;

gi(~), f'i('1) and hi(f3) denote the velocity patentials of' the i th distur-bances as illustrated in Fig. 6; and H [Ti} denotes the unit step f'unction

which equals zero prior to the arrival of' the i th wave (Ti

<

0) and equals unity af'ter the passage of' the fth wave f'ront (Ti> 0). The form of' the total velocity potentials CPl (Eq. 2.7) and

CP2

(Eq. 2.10) are quite

gener al but not arbi trary, as they have been chosen to satisfy both their respective wave equations (Eqs. 2.5 and 2.6) and their associated initial conditions (Eqs. 2.1 and 2.2 and Eqs. 2.3 ~~d 2.4). As an example of' satisfying the initial conditions the term -!:::'Pot/P2 in Eq. 2.10 accounts f'or the pressure dif'f'erence across the diaphragm (!:::'Po) f'or time t less than zero.

The presently unknown velo city potentials gi(~)' f'i('1) and hi«(3) can now be determined by evaluating the eff'ects of' two different boundaries on the wave motion . The f'irst boundary to be considered is the closed end of the driver. At this stationary boundary the particle velocity of the driver gas must be zero for all time. By setting the particle velocity

of the driver gas !:::,U2 (Eqs. 2.10 and 2.12) equal to zero at this boundary where distance x equals zero the following intermediate result can be obtained.

00

L

[fi(T}) H['1-2i+l} - hi«(3) H((3-2i+l}] = 0 i=l

(2.13)

The prime (I) denotes dif'ferentiation of' the variable with respect to the argument given in the f'ollowing brackets. Owing to the mathematical struc-ture of' this expression (Eq. 2.13) it will in general be identically zero f'or all time if ~~d only if' certain terms having equivalent step f'unctions cancel exactly. As '1 and (3 bath equal a2t/xo at the boundary x equals zero, the lengthy expression of' Eq. 2.13 can consequently be expressed in the f'ollowing equivalent and more convenient f'orm.

h!«(3) =f'!('1)

~ ~ i = 1, 2, ... , 00 (2.14)

On integrating each one of' these first-order differential equations with respect "to time, and af'ter setting the const~~ts of' integration equal to zero as they are arbitrary, the f'ollowing desired result ean be obtained.

i 1, 2, .'., 00 (2.15)

These results illustrate that the particle velo city of' the i th wave having a velocity potential f'i(T}) is countered exactly at the closed end of' the driver by the partiele velocity of the corresponding i th reflected wave having a veloeity potential hi «(3), thereby ha ving a net particle velocity of zero at this stationary boundary.

The second boundary to be considered is the interface of the driver and channel gases. At this contact surface the overpressure and particle velocity are taken to be continuous, and the effects of diffusion, heat transfer and turbulent mixing are neglected. The matching of the driver and

channel gas overpressures and particle velocities at the contact surface can be difficult mathematically because the contact surface is not stationary nor is its motion known a priori. To circumvent this difficulty the effects of the contact- surface motion are assumed to be negligible and the matching procedure is performed at the diaphragm station - a fixed or stationary location xo'

For the first step inthe matching procedure at the diaphragm station (x = xo) the overpressures of the channel gas 4>1 (Egs. 2.7 and 2.8) and the driver gas L.P2 (Eqs. 2.10 and 2.11) are equa'ted, yielding the following intermediate result.

00

alP1L[gi(s) H(s - 2(i-l)al /a2 + l}] =

i=l

00

-4> x _o

0

+ a2P2L[fi(T)) H(Tl-2i+l} + hi(t3) H(t3- 2i +l}] i=l

(2.16)

For this boundary (x = xo) s equals (alt - xo)/xo, I' equals (a2 t - xo)/xo and Tl equals (a2t + xo) /xo. Certain terms in Eq. 2.16 consequently have equivalent step functions, which can be grouped accordingly, such that Eg. 2.16 can be rewritten in 'the fOllowing equivalent and more convenient form.

i

=

1 (2.17)

i 2, 3,

0.

-

.,90

(2.18)

The first expression (Eg. 2.17) shows that at the contact surface the over-pressure of the first (shock) wave in the channel (gl) matches exactly the sum of the initial overpressure in the driver (L.po) and the overpressure of the first (rarefaction) wave in the driver (fl)' The second expression

(Eq. 2.18) shows that, for the i th wave interaction at the contact surface (see Fig. 6), the sum of the overpressures of the incident wave (hi-l) and reflected wave (fi) matches exactly the overpressure of the transmitted wave

(gi)·

The second and final step of the matching procedure at the diaphragm station (x

=

xo) is to eguate the particle velocity of the channel gas L.ul (Eqs. 2.7 and 2.9) to that of the driver gas L.u2 (Egs. 2.10 and 2.12), giving the following intermediate result.

, . - - - -- - - --

-00

I

[gi

(~) H(~

_{2(i-1)a1/a2 + lJ] =} i=l

00

I

[h{(~) H(~-2i+1}

- fi(T})

H(~

-

2i+1

}

]

(2.19) i=l

Like the previous result for the overpressure (Eq. 2.16) certain termsin Eq. 2.19 have equivalent step functions, which can be grouped accordingly, and Eq. 2.19 rewritten as shown below.

gi (~ ) = - f

i (

T) ) i =1

gi (~) = hi_1(~) - fi(T}) i=2,

3

, .•. ,

00

(2.20) (2.21)

The first expression (Eq. 2.20) shows that at the contact surface the par-ticle velocity of the first 6shock) wave in the channel (gl) matches exactly that of the first (rarefacti~n) wave in the driver (fl). The second ex-pression (Eq. 2.21) shows that, for the i th wave interaction at the contact

surface (see Fig. 6), the sum of the particle velocities of the incident wave (h_i _

1 ) and reflected wave (fi) matches exactly the particle velocity of the transmitted wave (gi).

There is actually a third boundary which is the end of the shock-tube channel. However, this boundary need not be considered here as it has already been implicitly assumed that the channel is either infinitely long or terminated by a perfect reflection eliminator such that no reflected or other waves arise from this boundary.

Now that the matching procedure has been completed the resulting equations can be used to obtain final expressions for the i th velocity potentials g;l(O, fi('I') and h· (~). From Eqs. 2.17 and 2.20, which are

simultaneous expressions in

git~)

and fi(T}), one can obtain giO) and fi(T}) expliëitly. In a similar manner, from Eqs. 2.18 ~~d 2.20, gi~~) and fi~T}) for i greater than unity can be obtained in terms of hi-l(~). These results are summarized below.

-Llp x gi(~) =

-

o 0 ~Pl _{+ a2P2} gi(~) (2a 2P2 = li.iiil °a: 1P1 + a2P2 Llp x fi(T}) - o 0 '8:₁Pl + a P _{2 2} fi(T}) =

~Pl

+ _a2

P

₂. ~Pl + a2P2 9 hi~l(~) h~- (~) ~-1 i=l (2.22) i=2, 3, •.. , 00 i=l i=2, 3, (2 .~3) 00

...

,

Then, in a straight-forward procedure using Eqs. 2~14, 2.22 and 2.23, explicit expressions can be obtained for g!(~), f!(Ti) and h!(~). These expressions can be easi1y integrated to yiêld the~final resmts for

g.(~), f.(Ti) and h.(~), which are given below.

~ ~ ~ -.6p x ~ gl (~)

=

~.., 0 0 , i=l , ~Pl + a 2 P2 (2.24) [: ~P1 - a2P2

r-

2 2a2 P2 .6p x ~ gi(~)

₌

0" o 0 i=2, 3, ...

,00

( alPl + a 2 P2 , .2 a p= + a 2P2 1 1

[

a1 P1 - a2P2

r-

l .6p x Ti f. (~)

=

o 0 i=l, 2, ••• ,00 . (2.25) ~ _~P1 + a 2P2 ~'alPl + a2P2

[

a1 P1 - ar P

r-

1 .6p x ~ h i (~) 2 '2 o 0 i=l, 2, ••• ,00 (2.26)

=

nc~ ? 8.J.P1 + a2P2 -él.1Pl + a2P2

Note that the constants of integration have all been set equal to zero .because they are arbitrary, and g!;(O, f!(Ti) and h~(~) can be easily

recov-ered by differentiation. Also, tÈese fiBal expres~ions for g.(~), f.(Ti) and h. (~) are valid away from the two boundaries x equal to zêro and~x as x in ~, Ti and ~ need no longer be restricted to zero or x • 0

o.

The solution for the wave motion in a rectangular shock tube util-izing different driver and channel gases has now been obtained. The final results for the i th velocity potentials g.(~), f.(Ti) and h.(~) (Eqs. 2.24, 2.25 and 2.26) can be substituted into thê expre~sions for~the total velocity potentials <1>1 and<l>2 (Eqs. 2.7 and 2.10). Then the two respective

expres-sions for the overpressure and particle velocity of the cliannel gas follow from Eqs. 2.8 and 2.9, andthose for the driver gas follow from Eqs. 2.11 and 2.12.

It can be seen from the solution of the wave motion in the rectan-gular shock tube that the specific impedances of the channel gas (~p )

and the driver gas (a

2P2) p1ay an important role. In fact, the important parameter is the ratio of these specific impedances. This ratio is denoted by the syIDbo1 ~ and it can be expressed in the following alternate forms by using the equation of state (p:=:=PRT) and the isentropic sound-speed rela-tion (a2 = /pIp· <=.: ,/,R T/M) •

o

(2.27)

The respective syrnbols ,/" p, R, T, R and M denote the specific heat ratio, pressure, gas constant, temperature,ouniversal gas constant (R_{o =} MR) and molecular weight. It is worth mentioning that the impedance ratio

a-

equals unity not only for the simplest case of identical driver and channel gases having the same temperatures but also for the speèific case of different

driver and channel gases provided that, ~/T l:quals, 2 ~/T2' Further-more, the impedanee ratio iS equals ('2

~/ï'

Mi)2 for the very important

case of differenr driver and channel gases

nav~ng equal temperatures, and

gequals (T

l/Tr )2 for the not so practicable case of identical driver ~~d

channel gases naVing different temperatur'es.

A srt of values of the specific impedanee ratio iS (~P2/alPl

=

('2 M21'2 _{M2)2 if Tl}=T_{2) was compiled for certain interesting different}

combinations of driver and channel gases having identical temperatures, and they are given in Table 2. The smallest values of 8 occur when a

light driver gas such as hydrogen (H2) or helium (He) is used in

conjunc-tion with a heavy channel gas such as dichlorodifluoromethane (CC12

F2),

sulfer hexafluoride (SF6) or octafluorocyclobutane (C4 F8)' On the other hand the large st values of iS occur when the driver gas has a high

molec-ular weight and the channel gas has a low molecmolec-ular weight. For nearly equivalent molecular weights of the driver and channel gases the specific impedanee ratio is always nearly equal to uni ty.

The wave that propagates in the channel gas is one of the most impor·t;ant features of the wave motion. For this reason the signature of pressure signature, as derived from Eqs. 2.7, 2.8, 2.24 and 2.27, is given below: llPl

=

IIp.o. H(

~

"I- l} 1+3

I

(X) [ 28 IIp (1 )i-2 al ] - ( 0)2 . - g H(§-- 2(i-l) -

+

l} (2/.28) ~+s ~+g ~ . i-2

After the arrival of the first shock wave (gl) at time (x-xo)/al at a fixed distanee x in the channel, each successive wave (gi) arrives at time (x-xo)/ ~ + 2(i-l)xo/~' or after equal time intervals of 2xo/~' Each successive

wave has an initial overpressure of zero and features a sudden change in overpressure at its wave front to a constant value thereafter. The first wave has an amplitude of llPo/(l+S), and each of the following waves has a

successively smaller amplitude by virtue of the factor (l-B)/(l+s) raised to the power i .. 2. Because each successive wave is superposed on the previous ones the total overpressure signature consists of an infinite sequence

of constant overpressure segments, each segment having a duration of 2xo/~'

The sum of terms giving each segment of the signature has the form of a finite ge0metric progression. Consequently the fOllowing geometrie sequence can be deri ved for the amplitude of the i th segment.

[1

_l

i-l IIp

-,8 0

~ 1 + S i=l, 2, •• ~,oo (2.29)

This expression is more convenient to use wh en constructing the overpressure signature than Eq. 2.28.

Five different predicted overpressure signatures of the wave moving in the channel are shown in Fig. 7, corresponding to specific impedanee ratios

tg)

of 0.2, 0.5, 1.0, 2.0 and 5.0. Although these impedanee ratios represent general cases of different driver and channel gases having different

tempera-tures, various important c"ombinations of' equivalent-temperature driver a.nd

channe1 gases which have such impedance ratios and would thus produce such signatures can be readi1y deduced from the inf'ormation given in Tab1e 2. The top two signatures in Fig. 7 for the case of' a light driver.:gas and a heavy channe1 gas (low

-

s

values of 0.2 and 0.5) exhibit a series of' very noticeab1e descending 'steps' and have re1ative1y high peak overpressures

(0.836P9_ and 0.67~po) f'or a given pressure dif'f'erence across the dia-phragm ~~Po). In general the duration of' each constant overpressure

segment (2xo/~) would be re1atively short because the sound speed of' a light or heated driver gas (a2) in relative1y high~(see the inf'ormation in Table 2 on the sound speed of dif'f'erent driver gases). The center

signature for the case when ~ has the value of' unity exhibits a single step having an amplitude of' 0.5L\po and a duration of' 2xo/a2. This signature

corresponds to the well-known case of' identical driver and channel gases having the same temperature • :~cl!owever, i t can also correspond to the

specif'ic case of different driver and channel gases provided 72 l-\/Tl

equals 72 M:2/T2. The bottom two signatures f'or the case of a heavy ari ver gas and a light channel gas (high values of' :B of' 2 and 5) exhibit a series of' steps that alternate markedly in amplitude f'rom positive to negative values as they diminish in absolute magnitude. The relatively low peak overpressure of' these two signatures (Oo33~po and O.17~o) in comparison to the pressure difference across the diaphragm (~Po) illustrates the

inef'f'iciency of' using a heavy driver gas to produce a relatively strong wave in a light channel gas. Note that these two signatures will have

rela-tive1y long durations because thersound speed of' the heavy driver gases will generally be low.

The particle-ve1oci ty signature of' the wave moving in the channel gas has the same features as the overpressure prof'i1e, because the partic1e ve10ci ty of' this planar wave is directly proportional to the overpressure

(L\u1 = ~P1/alP1)' The preceding remarks concerning the overpressure prof'i1e

theref'ore apply equally we11 to the partic1e-velocity signature.

The motion of the channel gas and contact surf'ace, owing to the wave motion in the rect~~ular shock tube, can be determined in the f'ol-lowing Illa.ILl1er. The partic1e-ve1oci ty signature like the overpressure prof'i1e consists of a sequence of descending or alternating constant over~

pressure steps or segments. The particle velocity (L\ul

=

~l/alPl)

associated with the i th segment of' its prof'i1e f'ollows f'rom Eq. 2.29 f'or the overpressure and iE given below.

1 [ 1-25

l

~ :::

ï+S

i =1,2, ••• ,00 (2.30)

During each constant partic1e-velocity segment

.of

the prof'ile, which has a f'ixed duration of' 2xo/~' the particle path wil1 be a 1inear f'unction of' time. The disp1acemënt of' a f'luid particle by the i th segment of' the wave is thus equal to the result of' Eq. 2.30 multiplied by 2xo/~' The accumu-lated displacement of' a f'luid partiele af'ter n successi ve segments of' duration 2x /~, or at successive times 2nxo/a2:measured f'rom the wave f'ront, can ge determined by addi ti on, and thi s re sult is shown below.

I[ik~.

n

=

1, 2, •••

,00

(2.31)

i=l

Because this sum is in the form of a geometrie progression, Eg.

2.31

can b~ easily expressed in an alternate a.11.d shorter form. By \!I'..oting that

a;

equals ,~P2/p2 or '2Pl(P2 to first order, the final expression for Ego

2.31

t~es tne fOlIoWing form.

n

= 1,2, •••

,00

(2.32)

In summary, this expression gives the displacement of a fluid partiele x-~ from its initial location ~ at successive times 2nxo/~ measured from the wave front. The displacement during each time interval 2xo/~ is a linear function of time.

The wave moving in the channel gas induces the same displacemert x-~ for each fluid particle because its predicted signature and amplitude are invariant with di stance • Consequently, the contact-surface displacenent x-x is also given by Eq. 2.32. The total displacement of any fluid

p~icle

and also the contact surface for large times

(n~

00) becomes in-dependent of the specific impedance ratio (s) and tends to a constant value of IIp Xol'2 Pl' This total displacement is normally small when compared with

~he

driver length 1xQ). For example, even if the pressure difference across the diaphragm. (llpo) is qui te high at Pl/20 (one twentieth of an atmosphere) and

,2

equals:_1.4 for a diatomic driver gas, the total dis-placement is only _{x o}/28 or 3.ff/o of the driver length. Note that the

original assUlI!Ption for the analysis that the contact-surface displacement was negligible compared with the d±±ver length was therefore very reasonable.

Three different particle paths corresponding to values of the specific impedance ratio s of 0.2, 1.0 and 5.0 are shown superposed on the time-distance diagram in Fig. 8. The displacement x-x* of a fluid particle fram its initial location x* has been exaggerated for clarity. For the case when s equals 0.2 the displacement increases monotomically to its maximum value of llpo x

I,

Plo This particle-path behav"iour is typical for the case when Blies ig

t~e

range from zero to unity, or when a light driver gas is used in conjunction with a heavy channel gas. For a driver and channel gas ha ving a specific impedance ratio of unity the displace-ment increases linearly to its maximum value IIp x

1'2

p in a time interval of

2xo/~,

after which the

displacemen~ i~

consta.11.t. For cases wh en the specific impedance ratio is greater than unity the displacement älternately increases and decreases linearly as the final value llpo xci

12

Pl.is approached, as illustrated in Fig.

8

by the particle path for whicli B equals

5.

The final displacement of the contact surface x-x , which equals llpo xOI'2 p , can be derived in an alternate manner, thereb~ providing a valuable inàependent check on the results of the preceding analysis for the rectanguJ.ar shock tube. Let the driver gas have an ini tial pressure of p + ~p and an initial density given by m

Ik!; ,

where m is the total mass or the °driver gas, Axo is the driver

vol~

eEd A is thg cross-sectional area of the shock .:tube. Now let the driver gas expand isentropically (as

it would for acoustic wave motion) to a final density of

mo/Ax

and a final pressure of Pl which is the initial pressure of the channel gas. The following isentropic expression relating the initial and final pressures and densities can therefore be appliedo

(2.33)

Consequently, to first order, the final ccontact-surface displacement x-xo equals 6po Xo/72-Pl, in exact agreement with the result of the previous acoustic analYsis.

A fUJ;'ther interesting and independent check on the acoustic analysis for the rectangular shock tube is provided by conventional shock-tube theory, which can be found in Ref. 17. For a constant-area shock tube utilizing perfect driver and channel gases having a smallor pressure difference across the diaphragm (6po)' the following conven-tional shock-tube equation can be used to predict the peak overpressure

(6Pl) of the constant-amplitude shock wave in the channel gas. -27

large

[ 1 + -6Po

J [

=

1 - 1 ; -6Pl

J [

l a}

..

(72-l)(~/a2H~Pl/Pl)

. . ] 7 2

-î

Pl Pl

.J

~

rl

(

_{71- 1 ) 6Pl!Pl}

+451

'.'

(2.34)

From this equation it can be easily shown toot, when 6p is much smaller that p , the peak overpressure of the first shock wave (.6J?l) equals

6Po/(1~

s), in exact agreement with the corresponding result of the previous acoustic analysis. It is worth mentioning that although the

conventional shock-tube equation is valid for large as well as small pressure differences across the diaphragm, the equation is limited in that i t can predict the amplitude of only the first but most important part of the wave in the channel gas. Admittedly, addi tional shock-tube theory can be applied to predict the complete wave motion for all time in the constant-area shock tube. However, the procedure is prohibitively difficult. Consequently, the acoustic solution for the complete wave motion in the shock tube, even though it is restricted t'o acoustic waves) is very valuable.

Some overpressure measurements were made in the channel of a constant-area shock tube, in order to provide an experimental check on the validity of the acoustic analysis for the rectangular shock tube. Three of the measured overpressure signatures are presented in Fig. 9. For the profile shown in Fig. 9a the driver and channel gases were both air having the same temperature. The profile shown in Fig. 9b corresponds to the case of helium in the driver and equivalent-temperature air in the

channel. For the opposite case of air in the driver and helium in the channel, the signature appears in Fig. 9c. Other pertinent data are given in the figure. Qualitatively the measurements substantiate the main features of the acoustic solution.

r - - - -- --_.

--For a quantitative comparison of measured and predicted overpres-sure signatures the meaoverpres-sured results of Fig. 9 have been reproduced in Fig. 10 and compared directly with the predicted signatures. Even for these

extremely strong acoustic"waves (~li tudes of about 5 kN/m2 or one-twentieth of au atIIlOsphere) the predicted and measured profiles are in good agreement. From experimental data such as those shown in Fig. 9 i t was found that the acoustic ~Dalysis was still valid when the pressure difference across the

diaphragm (6po) was as high as 25

KN/m2,

giving wave amplitudes of approximately 10 kN/:m2. Note that one of the main differences between measured and

pre-dicted signatures is that the rapid ch~Dges in overpressure in each experimentaJ. profile are not instantaneous (see Fig.

9).

The long ri se time of about 0.5 InS

associated with each of these shocks is due to poor diaphragm breakage result-ing from the relatively small pressure difference across the diaphragm.

3 .. PYRAMIDAL SHOCK TUBE 301 GBneral Solution

The pyramidal shock tube, which is depicted in Fig. 2b, consists essentially of both a pyramidal driver and channel having the same divergence angle and joined at a connnon-area station where a suitable diaphragm can be installed. This diaphragm separates a nOrmally atmospheric pressure channel gas from a slightly higher pressure driver gas. For generality of the

aneJ.ysis, wnich follows that in Chapter 2, let the il1.itially quiescent driver and channel gases be different and also have different temperatures. Before the diaphragm is broken the appropriate ini ti al condi tions C~D be summarized mathematically as follows 0 Channel (r

>

r ): o Driver (0

<

r

<

r ): o 6p ::: 0 1 6~ = 0 -6P2

=

6po 6U := 0 2 (3.1) (3.2) (3.3) (3.4)

The respecti ve symbols Lp, 6U, and 6p derrbte overpressure , particle velocity, and pressure difference across the di~phragm, whereas the respective subscri-pts 1 and 2 refer the appropriate symbols to the chanpel and driver gases. The radial distance r is measured along the shock tube starting from the apex of the pyramid as shown in Fig. 2b, and the diaphragm is located at a distance r , thereby making the driver length equal to r •

o 0

For the preceding initial conditions the ensuing wave motion in the pyramidal shock tube af ter the diaphragm is broken is rather complex. This wave motion can be depicted conveniently on a time-distance diagram, as shown in Fig.

6,

where the locus of each wave front has been drawn. On breaking the diaphragm in the shock tube the resulting rapidly expanding driver gas pro-duces a weak shock wave (g ) in the channel gas. Simultaneously a weak rare-faction wave (f

l ) moves into the driver gas and eventually reaches and re-flects from the driver apex. This reflected wave (~) propagates through

and channel gases. Because the contact-surface displacement is normal.ly negligible in comparison with the driver length, the contact-surface path is shown in Fig. 6 simply as avertical. dashed 1ine. Owing to the change in specific impedance (product of the density and sound speed) across the contact surface the reflected wave (b ) is partially transmitted to the channel

(g2) and partial.ly reflected tack into the driver (f

2). The subsequent osci11atory wave motion between the contact surface and driver apex creates a sequence of waves in the channeL The integrated result of al.l of these individual waves (g., i ::: 1, 2, ••• ,(0) yie1ds the totaJ. wave in the channel.

~

For a low-pressure-ratio pyramidal. shock tube for which the pressure difference across the diaphragm b.PQ is much less than the absolute pressure of the channel gas, the waves in the driver and channel are relatively weak. The wave motion can consequently be described' adequately by using wel1-known

acoustic theory. The one-dimensional. spherical. wave equations which govern the acoustic wave motion in the channel and driver gases are given below.

Channel (r

>

r ): o Driver (0

<

r

<

ro); 2 o (r<l>l)

~t

2

02 (r<l>2) ot2 ::: ::: 2 2 o (r<l>l) ~ _or2

(3.5)

2 2 ~ (r<l>2) a 2 _or2

(3.6)

The respective symbols <1>, a and t denote the total. velocity potential., sound speed and time.

For the analysis it is convenient to express the total velo city po-te.ntial. of the channel gas \~1) in its more basic form consisting of the sum

of velocity potentials of in~vidual. waves in the channel. This is simil-arly so for the total velocity potentiaJ. of the driver gas (<1>2). The new forms, and the related expressions for ovérpressure and particle velocity, are al.l summari~ed below.

Cha...rme1 t( r

>

r ):

o

00

<1>1

=

~

I

[gi (I;) H{t;-2i+2}

1

i=l 0<1>1 ::: -Pl

dt

_

~<I>i, -~ . (3.8)

Driver (0

<

r

<

r ):

o

()()

.6Po' t + _1 \ ' [ ]

~2 - - P2 r ~ fi(~) H(~-2i+l} + hi(~) H(~-2i+l}

i=l .6u = 2 ö~, P2

~t2

Ö~2

dr

(3.10) (3.12)

In the above expressions: P denotes density; the respective nondimensional

variables ~,

1)

and ~ equal (~t - r +:r )a

2/a,r ,(a2t+-r)/r and (a t - r)Jr ;

g!(~),

f!(T}) and

h!(~)

denote the

veloc~ty

potegtials of theOith

dis~urbance~

(~ee fig: 6) and tEe prime (t) denotes differentiation of the variable with respect to the argument given in the following brackets; and H(Ti} denotes the unit step function which equals zero prior to the arrival of the i th wave (Ti< 0) and equals unity af ter the arrival of its wave front (Ti>.b). The form of the total velo city potentials ~l and ~2 (Eqs. '3.7 and 3'~10) are quite general but not arbitrary, as they have been chosen to satisfy both their respective wave equations (Eqs. 3.5 and

3.6)

and their associated

ini-·tial conditions (Eqs. 3.1 and 3.2 and Eqs. 3.3 and 3.4). As an example of satisfying the initialr'conditions the term -.6po t/P2 in Eq. 3.10 accounts for the pressure difference across the diaphragm (.6Po) for time t less than zero. The presently known velocity potentials g!(~), f!(~) and h!(~) can now be determined by evaluating the effects of two aiffereftt boundarles on the wave motion. The first boundary to be considered is the driver apex. At

this stationary boundary the particle velo city of the driver gas must be zero for all time-. By setting the particle velocity of the driver gas ..6.~

(Eqs. 3010 and 3.12) equal to zero at this boundary where distance r is allowed to approach zero the following intermediate result can be obtained.

lim

r~O

()()

[~

I

(fi(~)

Hh -2i+l} -

hi(~) H(~-2i+~})

i=l

In the limit when r equals zero the following result can be obtained.

()()

I[f'i(1))

H(T} -2i+l} +

hi(~) H(.~

'

-2i+l}l

= 0 i=l

(3.13)

Owing to the mathematical structure of Eq. 3.14 this expression will in general be identically zero for all time if and only if certain terms having equivalent step functions cancel exactly. As T) and ~ both equal a

the boundary r equaJ.. to zero, the 1engthy Eq. 3.14 can consequentlI .. y be rewri tten in the fo11owing equivaJ..ent and more convenient form.

h! (~)

=

-f~ (Tj)

~ ~ i=l, '2, •••

,00

This result i11ustrates that the partic1e velocity of the ith wave having

(3.15)

a velocity potentiaJ.. f!(Tj) is countered exact1y at the driver apex by the partic1e velocity of tBe corresponding ref1ected wave having a velocity potentiaJ.. h!(~), thereby 1eaving no net partic1e velocity at the stationary

~

boundary.

The second Qoundary to be considered is the interface of the driver and channe1 gases. At the contact surface the overpressure and par-tic1e velocityare taken to be continuous after the diaphragm is removed, and the effects of diffusion, heat transfer and turbulent mixing are neg-1ected. The matching of the driver and channe1 gas overpressures and partic1e ve10cities at the contact surface can be difficult mathematicaJ..1y because the contact surface is not stationary nor is its motion known apriori. To circumvent this difficulty the effects of the contact-surface motion are assumed to be neg1igib1e and the matching procedure is done at the dia-phragm station - a fixed or stationary location r •

o

For the first step in the matching procedure at the diaphragm station (r=ro) the overpressure of the channe1 gas ~P1 (Eqs. 3.7 and 3.8) and the driver gas ÖP2 (Eqs. 3.10 and 3.11) are equated, thereby yie1d-ing the fo11owyie1d-ing intermediate result.

00

P1L[gi(~) H(~-2i-l-2}]=

-öpo

r~/a2

i=l

00

+ P2L[fi(Tj) H(Tj-2i+1} +

hi(~) H(~

-2i+1}] i=l

(3.16)

For the boundary in question for which r equaJ..s ro' ~ equaJ..s a₂t/r , Tj equaJ..s (a

2t + r )/r and ~ equaJ..s (a2t - r ) / r . Certain terms inOEq. 3.16 consequent~y h~ve equivaJ..ent step fun8tioRs, which can be grouped according1y, and expressed in the fo11owing equivaJ..ent and more convenient form.

i=l (3.17)

i=2,3, •••

,00.

(3.l8)

The first expression (Eq. 3.17) shows that at the contact surface the over-pressure of the first (shock) wave in the channe1 (gl) matches exact1y the sum of the initiaJ.. overpressure in the driver (öpo) and the overpressure of

r - - - -

-.--the first (rarefaction) wave in -.--the driver (fl ). The second expression (Eq. 3.18) shows that, for the i th wave interaction at the contact

surface (see Fig.

6),

the sum of the overpressures of the incident wave

(h

i_l) and reflected wave (fi ) ~tches exactly the overpressure of the

transmitted wave (gi).

Equation 3.17 and 3.18 can be integrated wi th respect to time to

yield the following useful results •

i=l (3.19)

= i=2, 3, •••

,00

(3.20)

The constants of integration are each equal to zero because the velocity

potentials g! (S), f! ('!)) and h! (~) are each equal to zero at their respective

wave fronts. J. Note that Eqs. J.3 •l9 and 3.20 illustrate that the product of

density and velocity potential (pct» is continuous at the contact surface

after the diaphragm has been removed, just like the overpresstire (.61') and

particle 'velocity (.6u).

The second and final step of the matching procedure at the dia-phragm station (r=r ) is to equate the particle velocity of the channel gas .6~ (Eqs. 3.7 and 3~9) with that of the driver gas .6~ (Eqs. 3.10 and 3.12), thereby giving the following intermediate result.

(3.21)

Like the previous result for the overpressure (Eq. 3.16) certain terms in Eq. 3.21 have equivalent step functions, which can be grouped accordingly, and expressed in the following equivalent and more useful form.

a

..3

g"(~)

+ gl'(n - -f"('!)) + f'('!)) ~ 1 1 1 i=l (3.22) a

~ g~'(S)

+

g~(~)

=

h~' {~)

+ 1r!

(~)

-

f~'('!))

+ f!('!)) ,°.0,

~:

• •• (3.23) ~ J.. 1 J.-l J.-l J. J. i=2,3, •••

,00

The first expression (Eq. 3.22) shows that at the contact surface the particle velo city of the first (shock) wave in the channel (gl) matches that of the first (rarefaction) wave in the driver (fl ). The second expression Eq. 3.23

the sum of the partic1e ve10cities of the incident wave (hi-1) f1ected wave (f~) matches exact1y the partic1e velo city of the mitted wave (g.).

J.

and re-

trans-There is actual1y a third boundary which is the large end of the pyralIlidal channe1. However, this boundary need not be considered hoere as it has already been imp1icit1y assumed that the channe1 is either in-finite1y long or terminated by a perfect ref1ection eliminator such that no reflected or other waves arise from this boundary.

Now that the matching procedure has been completed the resulting differential equations for g.(e), f.(~) and hi(~) can be cast into a more amendable form. From Eqs. 3:17, 3.i9 and 3.22 one can obtain differential equations for gl(Ç) and fl(~). In a simi1ar manner, from Eqs. 3.18 3.20 and 3.23, differential equations can be obtained for gi(Ç) and

fi(~)'

where i is greater than unity. These results associated with the boundary r

equal to ro and the previous result (Eq. 3.15) for the other boundary r

equal to zero are sumnarized below. 2 t.p r al o 0 (ç-1) alP 1 +a2P 2 a2 a1(P2-Pl) , g~(ç) + g (ç)

=

J. a:. P +a P i .L 1 2 2 i=l i=2, 3, ..• ,00 i=l a1(P2-P

i)

f~'(on) _J. + f' (on)

=

'I a p +a P i 'I 1 1 2 2 i=2, 3, .•• ,00 i=1;~2, ... ,00

The above set of differentia1 equations for g.(~), fi(TJ) and hi(~) can be uncoup1ed and subsequent1y solved to give ~xp1icit expressions for gi(O, fi(") and hi(~) .

(3.24)

(3.25)

(3.26)

A convenient initia1 step in solving the differentia1 equations is to find a recurrence re1ationship for f!(~). Using Eq. 3.26 we can rewrite Eq. 3.25 in the fol1owing form. J.

n-1

T

i

=

1

(3.27)

~

-1

ft! (-"-2)

+!

f~

(,,-2)

The nondimensional parameters iS and

i-

are equal to a2P2/alPl and (alPl+~P2)/al~P2-Pl) respectively. It should be noted that fi_l(Tj-2) and fi_ (Tj-2) in Eq. 3.27 are expressed correctly, as the variable Tj-2 in the

~racket

s (instead of simply Tj) is required to ensure that the func-tion f i _l (Tj-2) on the right-hand side of the equafunc-tion is properly phased in time with the function fi(Tj) on the left-hand side. Af ter multiplying the i th differential equation for fi(Tj) (Eq. 3.27) by exp [(Tj-2i+l)/i-] and partially integrating with respect to time the following recurrence relation for fi(Tj) can be obtained.

2'

.6p

r

'

i-

( 1 ) 1 (Tj-l)!

i-a 2P

:(so+ l)exp -

T

[(l+s+i-):'y + s] exp(y) dy

o

i=l (3.28)

s-l f' ( 2 ) 2 ( Tj-2i+l)J (Tj-2i+l)/t

s+i i-l Tj- t s+l c.exp - ] f! 1(ty+2i-3)

o ~- exp(y) dy

i=2, 3, ... ,00

From these results the ith velocity potential can be determined, and it is given below.

The notation (~) denotes the well-known binomial coefficients, which can be expressed às m~/n~(m-n) ~, where the symbol ~ denotes the factorial function. For convenience and illustrative purposes the first five velocity potentials

fi(~) are given in expanded form in Appendix A, along with the corresponding

f~rst four expanded functions I~(n.-2i+l). The functions I.(Tj-2i+l) can be

expressed in the following recu1rence form. J

J

w/i-{

eXP(_W/i-) [(l+s.+i-)y +'s] exp(y) dy j=l

I'.(w) =

oW/i-J exp(-wn)

J

Ij_l(y) exp(y) dy .j=2,3, ... ,00

,0

(3.30)

Alternatiire-ly the jth ftmction I'.(Tj-2i+l) can be expressed in the following

more elegant form. J

I~(w)

J

J

w/i-

I

w/i-=

exp( -wit) ... j... [(l+s+i-) y +

sJ

exp(y) dy

o 0

In eithe'r case the final results for the jth function I'.( Tj-2i+l) can be

determined, and it is given below. J

j Ij ( w ) = (1 +s +i-)

(ï -

.

j ) + s +

I

k=l k(l +S.+i-),-,s

(

r

w

)

j-k (j-k)

~

\

,

T

exp(-w/i-) 21 (3.31) (3.32)